# Thread: Convergence test for a series with e

1. ## Convergence test for a series with e

Hello,

I have to find out if this series converges:

$\sum_{n=1}^{\infty}{\frac{1}{n}\left(e^\frac{1}{n}-1\right)}$

Any help would be appreciated.

2. $e^x=1+x+O(x^2)$

So $\frac{1}{n}\left(e^{\frac 1n}-1\right)=\frac{1}{n}\left(1+\frac{1}{n}+O\left(\fr ac{1}{n^2}\right)-1\right)=\frac{1}{n^2}+O\left(\frac{1}{n^2}\right)$.

Now $\sum\left(\frac{1}{n^2}+O\left(\frac{1}{n^2}\right )\right)<\infty$, from which you can conclude that the original series also converged.

3. That solved my problem. Thank you very much for your help!